Friction, you make my head right round!

First of all this problem is not incorrect nor is it poorly stated.

Second,we do not need the values of mass and radius.

Now,considering the limiting case :

$1.$ On the sphere there will be two forces : $mg sin \theta$ <left> and ${\mu}_{2}mgcos \theta$ <right>

$2.$ On the plank there will be three forces : ${\mu}_{2}mgcos \theta$ <left> , $2mgsin \theta$ <left> and ${\mu}_{1}3mgcos \theta$ <right>

$3.$ Apply Newton's Second Law and you should get the following equations : ${a}_{sphere/earth} = gsin\theta - {\mu}_{2}gcos\theta$

${a}_{plank/earth} = \frac{g}{2}(cos\theta({\mu}_{2} - 3{\mu}_{1}) + 2sin\theta)$

From torque relation , $\tau = I\alpha$ < pure rolling,limiting case> $a = r\alpha$ >: we get ${a}_{sphere/plank} = \frac {5}{2}{\mu}_{2}gcos\theta$

Now, ${a}_{sphere/plank} = {a}_{sphere/earth} - {a}_{plank/earth}$

Equating, we get $\frac {{\mu}_{1}}{{\mu}_{2}} = \frac {8}{3}$ .